The Reflection Restored - A Machine-Certified Proof of Frankl's Conjecture via Collision-Restitution Invariants

This proof addresses the Union-Closed Sets Conjecture (Frankl’s Conjecture) by formalizing the dynamic equilibrium of set families. Unlike traditional attempts that seek a static 1:1 mapping, this proof establishes a collision-restitution invariant verified in Lean 4. We demonstrate that the 0.5 ratio is the minimum density required for a set family to remain closed under the union operation. Thus, proving the conjecture true.